Geodesic
Non-singular covers over ordered monoid rings
Mathematica Bohemica, Tome 131 (2006) no. 1, pp. 95-104

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MR Zbl
Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g h$ there is $l\in G$ such that $lg = h$. For a totally ordered cancellative monoid the equalities $Z(RG) = Z(R)G$ and $\sigma (RG) = \sigma (R)G$ hold, $\sigma $ being Goldie’s torsion theory.
Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g h$ there is $l\in G$ such that $lg = h$. For a totally ordered cancellative monoid the equalities $Z(RG) = Z(R)G$ and $\sigma (RG) = \sigma (R)G$ hold, $\sigma $ being Goldie’s torsion theory.
DOI : 10.21136/MB.2006.134079
Classification : 06F05, 16D50, 16D80, 16S36, 16S90, 18E40
Keywords: hereditary torsion theory; torsion theory of finite type; Goldie’s torsion theory; non-singular module; non-singular ring; monoid ring; precover class; cover class 
Bican, Ladislav. Non-singular covers over ordered monoid rings. Mathematica Bohemica, Tome 131 (2006) no. 1, pp. 95-104. doi: 10.21136/MB.2006.134079
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